I am looking for correlations between certain measures of brain structural integrity (fractional anisotropy, given as ratio between two hemisphere ==> rational data range 0-1, normally distributed) and behavioral parameters from stroke patients which are not normal but in part extremely leftskewed.
Having read numerous posts on assumptions of GLZM which I'd like to apply to assess the data, I still struggle with the assumption of non-normality of Y.
Normality assumption in linear regression ; http://www.talkstats.com/showthread.php/55824-Adjusting-non-parametric-correlation-for-covariates ; Assumptions of generalised linear model ; Assumptions for robust linear regression, specifically ANCOVA type models ; ANCOVA and its disturbing assumptions
For instance, it reads ".... In normal circumstances, the question isn't 'are my errors (or conditional distributions) normal?' - they won't be, we don't even need to check. A more relevant question is 'how badly does the degree of non-normality that's present impact my inferences?" I suggest a kernel density estimate or normal QQplot (plot of residuals vs normal scores). If the distribution looks reasonably normal, you have little to worry about. In fact, even when it's clearly non-normal it still may not matter very much, depending on what you want to do (normal prediction intervals really will rely on normality, for example, but many other things will tend to work at large sample sizes)...." I log transformed my dependent variable, can I use GLM normal distribution with LOG link function?
Moreover, here it says: ".... Strictly speaking, neither, though the second is the thing you check. What is assumed normal is either the unobservable errors, or equivalently the conditional distribution of Y at each combination of predictors. The unconditional distribution of Y is not assumed to be normal. – Glen_b May 30 '13 at 13:38 " Normality of dependent variable = normality of residuals?
or "....Since the residuals are just the y values minus the estimated mean (standardized residuals are also divided by an estimate of the standard error) then if the y values are normally distributed then the residuals are as well and the other way around. So when we talk about theory or assumptions it does not matter which we talk about because one implies the other." Normality of dependent variable = normality of residuals?
So my question is: Is a GLZM applied to my data valid if the residuals look normally distributed but the Y is not? Would a non-significant Shapiro-Wilk test on the residuals together with a reasonable QQ plots allow to trust the confidence intervals estimated incl. the p-vals?
I would be very grateful if somebody could help and clarify how to face this problem of non-normality in a generalized linear model. Having talked to two colleagues more familiar with modelling I got two different perspectives, A saying that normality of Y is crucial, B saying look at the residuals, GZLM should be robust to non-normal distributions.
Now I am confused with discrepant views on assumptions of GZLM.